If the angle between the vectors \( \vec{a}=2\lambda^2\hat{\imath}+4\lambda\hat{\jmath}+\hat{k} \) and \( \vec{b}=7\hat{\imath}-2\hat{\jmath}+\lambda\hat{k} \) is obtuse, then
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \lambda\in\left(0,\displaystyle\frac{1}{2}\right) \)
The angle is obtuse when \( \vec{a}\cdot\vec{b}\lt 0 \) (and the vectors are not in exactly opposite directions).
\( \displaystyle \vec{a}\cdot\vec{b}=14\lambda^2-8\lambda+\lambda \)
\( \displaystyle =14\lambda^2-7\lambda \)
\( \displaystyle =7\lambda(2\lambda-1) \)
\( \displaystyle 7\lambda(2\lambda-1)\lt 0\ \Rightarrow\ 0\lt\lambda\lt\frac{1}{2} \)
The vectors are never in opposite directions: \( \vec{a}=k\vec{b} \) would need \( 4\lambda=-2k \) and \( 1=k\lambda \), i.e. \( -2\lambda^2=1 \), which is impossible. So \( \lambda\in\left(0,\frac{1}{2}\right) \).
Why the other options are incorrect
Option (1) \( \lambda\in(0,\infty) \) — Incorrect. For \( \lambda\ge\frac{1}{2} \), e.g. λ = 1, \( \vec{a}\cdot\vec{b}=7\gt 0 \), so the angle is not obtuse.
Option (3) \( \lambda\in(-1,1) \) — Incorrect. For λ = −0.5, \( \vec{a}\cdot\vec{b}=3.5+3.5=7\gt 0 \), so the angle is not obtuse.
Option (4) \( \lambda\in(-\infty,0)\cup\left(\displaystyle\frac{1}{2},\infty\right) \) — Incorrect. This is where \( \vec{a}\cdot\vec{b}\gt 0 \), i.e. where the angle is acute.